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Suppose $Ax^2+Bxy+Cy^2+Dx+Ey+k=0$ is a conic in the Euclidean plane. How do I recognize what is it? In my book they have proved the determinant test that if $B^2-4AC$ is $>0$ if hyperbola, $=0$ if parabola and $<0$ if it is an ellipse.

But my confusion is that they do not include pair of straight lines and the circle(though it is a special case of the ellipse).

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Is this what you want?

Let $p=B^2-4AC$.

If $p\lt 0$, ellipse, circle, point or no curve.

If $p=0$, parabola, 2 parallel lines, 1 line or no curve.

If $p\gt 0$, hyperbola or 2 intersecting lines.

more information here with figures.

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$1. $ if it is a circle $$A=C$$ and $B=0$

$2.$ if it is a pair of straight line $$B^2-4AC\ge 0$$ and $$ACK+\frac{BDE}{4}-\frac{AE^2}{4}-\frac{CD^2}{4}-\frac{KB^2}{4}=0$$

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